(29 January 1810 – 14 May 1893)
See as well some earlier references to diamond signs here .
The proper context for some diamond figures that I am interested in
is the 4×4 array that appears, notably, in Hudson's 1905 classic
Kummer's Quartic Surface . Hence this post's "Kummerhenge" tag,
suggested also by some monumental stonework at Tufte's site.
The Hudson array mentioned above is as follows —
See also Whitehead and the
Relativity Problem (Sept. 22).
For coordinatization of a 4×4
array, see a note from 1986
in the Feb. 26 post Citation.
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24 with the aid of the MOG.
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
The 15 2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)
Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.
See also Correspondences in this journal.
* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .
Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:
See as well the word correspondences in
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).
From "Six Significant Landscapes," by Wallace Stevens (1916) —
VI
Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses —
As, for example, the ellipse of the half-moon —
Rationalists would wear sombreros.
But see "cones, waving lines, ellipses" in Kummer's Quartic Surface
(by R. W. H. T. Hudson, Cambridge University Press, 1905) and their
intimate connection with the geometry of the 4×4 square.
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies's "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K6 and to the 15 2-subsets of a 6-set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2-subsets of a 6-set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6-set
in the work of other authors in 1994 and 2002 . . .
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
|
http://www.math.sci.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt
Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve. |
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
A Buddhist view —
“Just fancy a scale model of Being
made out of string and cardboard.”
— Nanavira Thera, 1 October 1957,
on a model of Kummer’s Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
The above four-element sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements. This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
Related material —
Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge and Nanavira Thera.
See as well Bill Murray's 1984 film "The Razor's Edge" …
Movie poster from 1984 —
"A thin line separates
love from hate,
success from failure,
life from death."
Three other dualities, from Nanavira Thera in 1959 —
"I find that there are, in every situation,
three independent dualities…."
(Click to enlarge.)
The previous post discussed the parametrization of
the 4×4 array as a vector 4-space over the 2-element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Some context for yesterday's post on a symplectic polarity —
This 1986 note may or may not have inspired some remarks
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .
See also the diamond-theorem correlation.
The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.
Example from mathematics:
Click the Barth passage to see it with its surrounding text.
Related material:
Mathematical
Literary (also from May 18, 2010)
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
|
Narrative:
Aooo.
Happy birthday to Stephen King.
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space." The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."
The following excerpt from a January 20, 2013, preprint shows that
a Galois-geometry version of the large Desargues 154203 configuration,
although based on the nineteenth-century work of Galois* and of Fano,**
may at times have twenty-first-century applications.
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 154203 configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428-435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106-132.
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6-volume
Principles of Geometry .
Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
Background: Rosenhain and Göpel Tetrads in PG(3,2)
|
Introduction: The Large Desargues Configuration Added by Steven H. Cullinane on Friday, April 19, 2013 Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point The converse, from the same book:
"If two triangles are perspective from a line
Desargues' theorem according to Wikipedia
"Two triangles are in perspective axially [i.e., from a line]
A figure often used to illustrate the theorem,
A discussion of the "if and only if" version of the theorem
This large Desargues configuration involves a third triangle,
Point-line incidence in this larger configuration is,
The third triangle, within the larger configuration,
|
A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desargues-theorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original Veblen-Young figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.
In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):
|
Volume I of Baker's Principles has a cover closely related to the Rosenhain tetrads in PG(3,2) |
Volume IV of Baker's Principles has a cover closely related to the Göpel tetrads in PG(3,2) |
|
Foundations (click to enlarge)
|
Higher Geometry (click to enlarge)
|
On this date 106 years ago…
Prefatory note from Hudson's classic Kummer's Quartic Surface ,
Cambridge University Press, 1905—
RONALD WILLIAM HENRY TURNBULL HUDSON would have
been twenty-nine years old in July of this year; educated at
St Paul's School, London, and at St John's College, Cambridge,
he obtained the highest honours in the public examinations of the
University, in 1898, 1899, 1900; was elected a Fellow of St John's
College in 1900; became a Lecturer in Mathematics at University
College, Liverpool, in 1902; was D.Sc. in the University of London
in 1903; and died, as the result of a fall while climbing in Wales,
in the early autumn of 1904….
A many-sided theory such as that of this volume is
generally to be won only by the work of many lives;
one who held so firmly the faith that the time is well spent
could ill be spared.
— H. F. Baker, 27 March 1905
For some more recent remarks related to the theory, see
Defining Configurations and its updates, March 20-27, 2011.
The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.
From that article:
The following corrects the word "unique" in the example.
* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.
_____________________________________________________________
Update of March 21
The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—
In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.
Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.
Update of March 22 (itself updated on March 25)
For further background on configurations, see Dolgachev—
Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.
Update of March 27
See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.
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